Exploring Integer Multiplication
Focus on…
After this lesson, you
will be able to...
φ multiply integers
using integer
chips
The Columbia Icefield is the largest mass of ice in North America below
The Columbia
Icefield is a major
source of fresh water.
Melt water from the
icefield feeds rivers
that flow to the
Arctic Ocean, the
Pacific Ocean, and
Hudson’s Bay.
the Arctic Circle. The icefield lies across the Alberta–British Columbia
border in the Rockies. Six large glaciers flow from the icefield. One of
them, the Athabasca Glacier, is a popular tourist destination in Jasper
National Park.
The Athabasca Glacier has been melting for over a century. The front edge
or “snout” of the glacier has been receding at an average of approximately
12 m per year. At that rate, how far would it recede in four years?
How can you use integer chips to multiply two integers?
• red and blue integer
chips
Literacy
Link
Representing
Integers
Integer chips are
coloured disks that
represent integers. In
this book, one red
chip represents +1,
and one blue chip
represents -1.
+1
286
−1
1.
Multiplication can be expressed as a repeated addition.
(+3) × (+2) = (+2) + (+2) + (+2)
a) Use red integer chips to model the addition (+2) + (+2) + (+2).
b) Copy and complete the multiplication statement (+3) × (+2) = .
Express (+4) × (+3) as a repeated addition.
b) Use red integer chips to model the addition.
c) Copy and complete the multiplication statement (+4) × (+3) = .
2. a)
Express (+3) × (-5) as a repeated addition.
b) Use blue integer chips to model the addition.
c) Copy and complete the multiplication statement (+3) × (-5) = .
3. a)
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4.
5.
Use integer chips to model each multiplication. Copy and complete
each multiplication statement.
a) (+3) × (+5) = b) (+2) × (+3) = c) (+4) × (-3) = d) (+3) × (-4) = Can you use the same method as in #1 to #4 to complete the
multiplication (-2) × (+3) or the multiplication (-3) × (-4)?
Explain.
6. a)
The diagrams show how to model the multiplications from #5
by using zero pairs. Describe each model.
(-2) × (+3)
Use enough zero pairs so that there
are two groups of three red chips
available to remove.
Literacy
Understanding
Multiplication
The product of 4
and 2 is 8, because
4 × 2 = 8.
The multiplication
statement 4 × 2 = 8
means that 4 groups
of 2 make 8. You can
also think of 4 × 2 as
the repeated addition
2 + 2 + 2 + 2.
Literacy
(-3) × (-4)
Link
Link
Modelling With
Zero Pairs
A zero pair is a pair of
integer chips, with
one chip representing
+1 and one chip
representing -1.
+1
−1
A zero pair represents
zero because
(+1) + (-1) = 0.
Any whole number
of zero pairs
represents zero.
b)
7.
Copy and complete each multiplication statement.
(-2) × (+3) = (-3) × (-4) = Use integer chips to model each multiplication. Copy and complete
each multiplication statement.
a) (-4) × (-3) = b) (-5) × (+2) = c) (-2) × (+4) = d) (-1) × (-4) = Reflect on Your Findings
8.
How can you use integer chips to multiply two integers? In your
description, state when you use zero pairs. How do you determine
the number of zero pairs to use?
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Example 1: Multiply Using Integer Chips
Determine each product using integer chips. Copy and complete each
multiplication statement.
a) (+5) × (+2)
b) (+6) × (-2)
c) (-3) × (+2)
d) (-2) × (-4)
Solution
Strategies
a)
Model It
There are 10 red
chips.
Strategies
(+5) × (+2) = (+2) + (+2) + (+2) + (+2) + (+2)
Insert 5 groups with 2 red chips in each group.
What other strategy
could you use?
The product is +10.
The multiplication statement is (+5) × (+2) = +10.
b)
There are 12
blue chips.
(+6) × (-2) = (-2) + (-2) + (-2) + (-2) + (-2) + (-2)
Insert 6 groups with 2 blue chips in each group.
The product is -12.
The multiplication statement is (+6) × (-2) = -12.
c)
The negative sign in -3 shows that you need to remove 3 groups.
Each group will contain 2 red chips. So, you need to remove a total
of 6 red chips.
Start with 6 zero pairs, so
that there are enough red
chips to remove.
Now remove the
red chips.
There are 6 blue
chips.
The product is -6.
The multiplication statement is (-3) × (+2) = -6.
288
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d)
The negative sign in -2 shows that you need to remove 2 groups. Each group
will contain 4 blue chips. So, you need to remove a total of 8 blue chips.
Start with 8 zero pairs, so
that there are enough blue
chips to remove.
Now remove
the blue chips.
There are
8 red chips.
The product is +8.
The multiplication statement is (-2) × (-4) = +8.
Determine each product using integer chips. Use diagrams to show
your thinking.
a) (+4) × (+2) b) (+5) × (-2) c) (-4) × (+2) d) (-6) × (-1)
Example 2: Apply Integer Multiplication
For 5 h, the temperature in Flin Flon fell by 3 °C each hour. What
was the total change in temperature?
Solution
Use the multiplication of two integers to represent the situation.
Represent the time of 5 h by the integer +5.
Represent the 3 °C decrease in each hour by the integer -3.
The total change in temperature can be represented
by the expression (+5) × (-3).
Multiply (+5) × (-3) using integer chips.
The number of hours
times the change per
hour gives the total
change.
The product is -15.
The total change in temperature was a decrease of 15 °C.
For 4 h, the temperature in Victoria fell by 2 °C each hour.
What was the total change in temperature?
Flin Flon lies along
the Manitoba–
Saskatchewan
border. The town is
named after a
fictional character
called Professor
Josiah Flintabbatey
Flonatin. He was the
hero of a science
fiction novel called
The Sunless City. In
this novel, Josiah
explored a
bottomless lake in
a submarine and
discovered a tunnel
lined with gold. Flin
Flon was named
after him because
of the large mineral
deposits discovered
there.
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• To model the multiplication of an integer by a positive integer, you can
insert integer chips of the appropriate colour.
(+2) × (-3) = -6
• To model the multiplication of an integer by a negative integer, you
can remove integer chips of the appropriate colour from zero pairs.
(-2) × (-3) = +6
1.
David said that he could model the multiplication (+3) × (-7)
using 3 red chips and 7 blue chips.
a) Do you agree with David? Explain.
b) What chips would you use to model (+3) × (-7)? Explain.
2.
To model (-3) × (-5), Raini places
15 zero pairs on her desk.
a) Why did she use 15 zero pairs?
b) What should she do next?
3. a)
b)
4.
290
Paolo models (-2) × (+3) as shown in the diagram. He
determines the correct product, -6. Explain why his method works.
Could Paolo model the product if he started with 4 zero pairs? Explain.
Jasmine said that she did not need any integer chips to multiply
0 × (+2) or to multiply (-3) × 0. Explain her thinking.
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10.
What multiplication statement does
each set of diagrams represent?
a)
For help with #5 to #14, refer to Example 1 on
pages 288–289.
5.
Write each repeated addition as a
multiplication.
a) (+1) + (+1) + (+1) + (+1) + (+1)
b) (-6) + (-6)
6.
Write each expression as a multiplication.
a) (+7) + (+7) + (+7)
b) (-4) + (-4) + (-4) + (-4)
7.
Write each multiplication as a repeated
addition.
a) (+3) × (+8)
b) (+5) × (-6)
8.
Write each expression as a repeated
addition.
a) (+7) × (+2)
b) (+4) × (-9)
9.
What multiplication statement does
each set of diagrams represent?
b)
11.
What multiplication statement does
each set of diagrams represent?
a)
a)
b)
b)
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12.
What multiplication statement does each
set of diagrams represent?
15.
An aircraft descends at 3 m/s for 12 s.
Use the multiplication of two integers to
represent the situation. How far does the
aircraft descend?
16.
A building has 10 storeys above ground
and 3 storeys below ground. Each storey
has a height of 4 m.
a) What is the total height of the building
above ground?
b) What is the total depth of the building
below ground?
17.
An oil rig is drilling a well at 2 m/min.
How deep is the well after the first 8 min?
18.
Does doubling an integer always result
in an integer of greater value? Explain.
19.
In a magic square, the numbers in each
row, column, and diagonal have the same
sum. This is called the magic sum.
a) What is the magic
+2
+3
-2
sum for this magic
square?
a)
b)
13.
Copy and complete each multiplication
statement.
a) (+4) × (+6)
b) (+7) × (-2)
c) (-1) × (+5)
d) (-8) × (-2)
-3
+1
+5
+4
-1
0
Multiply each integer in the square by
-2. Is the result another magic square?
If so, what is the new magic sum?
c) Create a magic square with a magic
sum of -12.
b)
For help with #14 to #17, refer to Example 2 on
page 289.
14.
292
Use the multiplication of two integers to
represent each situation. Then determine
the product and explain its meaning.
a) The temperature increased for 6 h at
2 °C per hour.
b) Ayesha repaid some money she owed
in 4 instalments of $8 each.
20.
Arrange the following numbers of +1s
and -1s in the small squares on a threeby-three grid so that each row, column,
and diagonal has a product of -1.
a) six +1s, three -1s
b) four +1s, five -1s
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